Simplify and expand the following expression: $ \dfrac{4}{a + 4}- \dfrac{3}{a - 3}- \dfrac{3a}{a^2 + a - 12} $
Answer: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor the quadratic in the third term: $ \dfrac{3a}{a^2 + a - 12} = \dfrac{3a}{(a + 4)(a - 3)}$ Now we have: $ \dfrac{4}{a + 4}- \dfrac{3}{a - 3}- \dfrac{3a}{(a + 4)(a - 3)} $ The least common multiple of the denominators is: $ (a + 4)(a - 3)$ In order to get the first term over $(a + 4)(a - 3)$ , multiply by $\dfrac{a - 3}{a - 3}$ $ \dfrac{4}{a + 4} \times \dfrac{a - 3}{a - 3} = \dfrac{4(a - 3)}{(a + 4)(a - 3)} $ In order to get the second term over $(a + 4)(a - 3)$ , multiply by $\dfrac{a + 4}{a + 4}$ $ \dfrac{3}{a - 3} \times \dfrac{a + 4}{a + 4} = \dfrac{3(a + 4)}{(a + 4)(a - 3)} $ Now we have: $ \dfrac{4(a - 3)}{(a + 4)(a - 3)} - \dfrac{3(a + 4)}{(a + 4)(a - 3)} - \dfrac{3a}{(a + 4)(a - 3)} $ $ = \dfrac{ 4(a - 3) - 3(a + 4) - 3a} {(a + 4)(a - 3)} $ Expand: $ = \dfrac{4a - 12 - 3a - 12 - 3a}{a^2 + a - 12} $ $ = \dfrac{-2a - 24}{a^2 + a - 12}$